construct the $180^{\circ}$ rotation of the square abcd about point p.

choose the correct figure of the rotation below.

Question

construct the $180^{\circ}$ rotation of the square abcd about point p.

choose the correct figure of the rotation below.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Rigid Motion", "Geometric Transformations" ], "new_concepts": [ "Rotations in the Coordinate Plane" ], "current_concepts": [ "Rigid Motion", "Geometric Transformations", "Rotations in the Coordinate Plane" ] } ## Identify the coordinates of the original square and the center of rotation $$ \begin{aligned} &P = (3, 3)\ &A = (7, 4),\quad B = (8, 4),\quad C = (8, 3),\quad D = (7, 3) \end{aligned} $$ ## Apply the $180^\circ$ rotation formula about point $P(x_0, y_0)$ $$ \begin{aligned} &(x', y') = (2x_0 - x, 2y_0 - y) = (6 - x, 6 - y)\ &A' = (6 - 7, 6 - 4) = (-1, 2)\ &B' = (6 - 8, 6 - 4) = (-2, 2)\ &C' = (6 - 8, 6 - 3) = (-2, 3)\ &D' = (6 - 7, 6 - 3) = (-1, 3) \end{aligned} $$ ## Match the calculated coordinates with the given options $$ \begin{aligned} &\text{Option B shows the rotated square with vertices:}\ &A' = (-1, 2),\quad B' = (-2, 2),\quad C' = (-2, 3),\quad D' = (-1, 3) \end{aligned} $$

(A) Rotated square with vertices $A'$ at $(-1, 1)$, $B'$ at $(-2, 1)$, $C'$ at $(-2, 2)$, $D'$ at $(-1, 2)$

(B) Rotated square with vertices $A'$ at $(-1, 2)$, $B'$ at $(-2, 2)$, $C'$ at $(-2, 3)$, $D'$ at $(-1, 3)$

(C) Rotated square with vertices $A'$ at $(1, 1)$, $B'$ at $(2, 1)$, $C'$ at $(2, 0)$, $D'$ at $(1, 0)$

{ "elements": [ { "type": "point", "params": [[3, 3]], "properties": { "name": "P", "size": 4, "color": "#F2557F", "withLabel": true } }, { "type": "polygon", "params": [[7, 4], [8, 4], [8, 3], [7, 3]], "properties": { "fillColor": "rgba(140, 85, 242, 0.3)", "strokeColor": "#8C55F2", "strokeWidth": 2, "name": "ABCD" } }, { "type": "polygon", "params": [[-1, 2], [-2, 2], [-2, 3], [-1, 3]], "properties": { "fillColor": "rgba(85, 131, 242, 0.3)", "strokeColor": "#5583F2", "strokeWidth": 2, "name": "A'B'C'D'" } }, { "type": "point", "params": [[7, 4]], "properties": { "name": "A", "color": "#8C55F2" } }, { "type": "point", "params": [[-1, 2]], "properties": { "name": "A'", "color": "#5583F2" } } ] } { "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Rotations in the Coordinate Plane" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the coordinates of the original square and the center of rotation

Apply the $180^\circ$ rotation formula about point $P(x_0, y_0)$

Match the calculated coordinates with the given options

Answer: